The spectral properties of the adjacency (connectivity) and distance matrix for various types of networks: exponential, scale-free (Albert–Barabási) and classical random ones (Erdős–Rényi) are evaluated. The graph spectra for dense graph in the Erdős–Rényi model are derived analytically.

The Erdős-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph mod...

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdős-Rényi graphs. We investigate further properties of our bounds, and show how our results on the Erdős-Rényi graphs can be extended to other polarity graphs.

Distributions of abundances or frequencies play an important role in many fields of science, from biology to sociology, as does the Rényi entropy, which measures the diversity of a statistical ensemble. We derive a mathematical relation between the abundance distribution and the Rényi entropy, by analogy with the equivalence of ensembles in thermodynamics. The abundance distribution is mapped o...

Consider the binary classification problem of predicting a target variable Y from a discrete feature vector X = (X1, . . . , Xd). When the probability distribution P(X, Y ) is known, the optimal classifier, leading to the minimum misclassification rate, is given by the Maximum A-posteriori Probability (MAP) decision rule. However, in practice, estimating the complete joint distribution P(X, Y )...

Rényi entropies Sq are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q ≥ 0. For (d + 1)dimensional conformal field theories, the Rényi entropies across S may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R×H, where H is the d-dimensional hyperb...